Hi megalizardfish,
You should be able to find the serial number in 2 locations:
1) On the top of the retail box in the white box (as Penanc3 noted)
2) On the backplate of the card, between the screws of the heatsink
Please let me know if you have any further questions, or feel free to contact our support team directly at 1-877-596-6822 9:00am – 6:00pm PST Mon – Fri or send an email to support.us@zotac.com
Sincerely,
ZOTAC